A sharp upper bound for the rainbow 2-connection number of 2-connected graphs

TL;DR

This paper establishes that for all 2-connected graphs, the rainbow 2-connection number is at most equal to the number of vertices, with equality only for cycles, thus solving a previously posed problem.

Contribution

The paper proves the exact upper bound of the rainbow 2-connection number for 2-connected graphs and characterizes when equality holds.

Findings

For all 2-connected graphs, rc_2(G) ≤ n.

rc_2(G) = n if and only if G is a cycle.

Confirmed the minimal constant α = 1 for the bound.

Abstract

A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is defined to be the minimum number of colors required to color the edges of $G$ such that every two distinct vertices of $G$ are connected by at least $k$ internally disjoint rainbow paths. Fujita et. al. proposed a problem that what is the minimum constant $\alpha>0$ such that for all 2-connected graphs $G$ on $n$ vertices, we have $rc_2(G)\leq \alpha n$. In this paper, we prove that $\alpha=1$ and $rc_2(G)=n$ if and only if $G$ is a cycle of order $n$, settling down this problem.